Question on increasing airflow to a room that does not a mini split head unit.

Basic setup of house – standard two story – open floor plan on first floor – one head unit located near one side of house (northside – not central). Second floor has head unit in master bedroom, also located on north side of house. 

These were sized correctly, getting great efficiency. The only issue is comfort in the upstairs bedrooms on south side of house (where no head units are located) in the middle of summer. These rooms are not used all the time, but it is a dead heat zone in the middle of summer – very little airflow. 

I have tried to run a fan down the hallway to push air from upstairs head unit but does not drop temperature much at all. 

I was thinking of drilling a vent with fan through an upstairs bedroom floor into the downstairs dining room (remember this is all on south side – all head units are on north side). 

My idea is to either draw the heat out of one bedroom or push cold air into the bedroom from the first floor. I would turn the fan on when room was in use. 

First question – will this work/help? Also, is it better to draw the heat out or push the cool air in (fan direction)? The door to the bedroom will be open. I am worried about drawing hot air in through leaks in windows/attic by creating too much pressure difference. 

Let me know if more detail is needed. Thanks!!!